Conventional piezoelectric materials produces a voltage under stress (the piezoelectric effect), and deform under an applied electric field (the converse piezoelectric effect). No material has ever been produced that shows the piezoelectric effect without also having the inverse piezoelectric effect, as the direct and converse effects are thermodynamically identical.
Further, the conventional belief is that piezoelectric materials must be non-centrosymmetric, or at least contain a non-centrosymmetric component, which severely limits the material choices available. The most commonly used piezoelectric material is lead zirconate titanate, but there are environmental and public health concerns related to the production and use of any lead-containing material. It has proved difficult to find any better material, using conventional approaches.
Piezoelectric devices have many useful applications, such as high voltage generation (e.g. gas lighters using the resulting spark), microactuators, microbalances, acoustic generators (including ultrasound generators), vibration sensors, and the like. It is impossible in conventional piezoelectrics to break the connection between direct and converse effects. It is also difficult to make either thick or thin film piezoelectrics of high sensitivity. Most current piezoceramics are based on lead containing perovskite structure compositions, and as noted above this is less than ideal. Applications would increase if improved materials were available.
The flexoelectric effect relates to an electric polarization induced by a strain gradient within a material, and the converse effect is a strain in the material induced by an electric field gradient. A flexoelectric material can be centrosymmetric, which would seem to rule out any piezoelectric effect.
The flexoelectric effect is defined by the relationship:
                              P          1                =                              μ                          ijk              ⁢                                                          ⁢              1                                ⁡                      (                                          ∂                                  S                  ij                                                            ∂                                  x                  k                                                      )                                              (        1        )            where μijkl are the fourth rank polar tensor flexoelectric coefficients,
Sij is the elastic strain components.
Xk is the direction of the gradient in S, and
Pl is the induced electric polarization.
For flexoelectricity there is also a converse effect, i.e. there is an elastic stress generated by an electric field gradient defined by the relationship:
                              T          ij                =                              μ                          ijk              ⁢                                                          ⁢              1                                ⁡                      (                                          ∂                                  E                  k                                                            ∂                                  x                  1                                                      )                                              (        2        )            where Ek is the electric field,
xl the direction of the gradient in E, and
Tij the induced stress.
For the direct effect in the MKS system, units for μ are coulombs/meter (C/m), and for the converse effect the units are Newton/volt (N/V), which are necessarily equivalent as the direct and converse flexoelectric effects are thermodynamically identical.